kitchen table math, the sequel: 12/31/06 - 1/7/07

Saturday, January 6, 2007

helicopter parents pressuring their kids

Looks like another story that will bash "helicopter parents" who are too involved in their children’s education.

"One expert told me (for a story I’m working on for InTown: Westchester on Helicopter Parents) that the pressure to succeed begins in elementary school. Parents are using tutors for their elementary school kids not for remedial help, but for enrichment and to get them ahead, says Lisa Jacobson of Inspirica, a tutoring and test prep service based in Manhattan ( Ahead at age six! What’s happening to this world?"

So Little Time

Yeah, I’m looking to "enrich" my daughter so she knows her times tables in fourth grade!

Friday, January 5, 2007

a sorry state of affairs

Here's a nice little letter to the editor concerning the sorry state of math instruction:

I'd like to address the MCAS test and math education together. If what I suggest is happening (inflated evaluations in our schools) is true, we do need to have some measuring instrument which will accurately assess the comprehension of material beyond what the schools are reporting. In 1970, 6 percent of college freshmen reported having an "A" average in high school. In 2005, this figure was over 22 percent. In 1970, most of my 11-year-old sixth graders could pass a test on fractions, decimals and percents. In 2004, many of my above-average freshmen struggled if I included those concepts on an algebra test. Recently, a college-age clerk at a food store asked if 99 cents a pound was "close enough" to "a little over a half a pound." A junior college student thought that there were 12 yards in a mile. Understanding of this degree, or should I say misunderstanding, is just not acceptable.

That does seem to comport with today's math reality, rather than the rosy rhetoric. Math is supposedly now being taught with understanding, yet what we see is that many students have little real understanding of math and little facility solving simple math problems.

The traditional curriculum was far from perfect--too few kids learned higher math skills. But. today's math instruction seems to have gone from mediocre to worse.

I, for one, blame NCTM who are largely responsible for this current state of affairs. They claim to know how to teach math to kids. In reality, they don't. Their bromides have been failures. Sure, you can go through their standards and twist the words to come up with some sound math principles, but, overall, the framework they've laid down has resulted in actual instruction that is worse than what we had before.

It's time to give them the boot.

weighted GPA

Our High School is proposing to change the way high school grade point average is calculated. We currently use a a weighted GPA scale that gives more "credit" to Honors and AP level courses. Honors and AP are graded on a 6.25 scale (an A+ gets a 6.25) and the non-Honors/AP courses use a 5.25 scale. Also, health and physical ed courses are not included in the GPA calculation.

The high school administration proposes to end this and grade all students on a single 4.0 scale that includes all graded courses (scooping in band, pe, health, maybe others). I'm on the fence on this because I really don't know what the pros and cons are and I am hoping maybe some of you out there might help clarify.

Here's what we've come up with so far -- the change would primarily affect the class "rank." By taking easier courses and getting an A, you'll get a better GPA and a better class rank than if you take AP and Honors courses and get a B. The kids graduating in that all important (is it?) top 10% might be those that avoid AP and Honors and take easier courses for a better grade. Would this new system penalize kids that stretch themselves in AP and Honors? Or do colleges really not care about GPA and class rank, caring much more about the courses on the transcript rather than the grades achieved?

Thursday, January 4, 2007

help desk

I worked an "overlapping triangle" problem in Saxon Algebra 2 today that threw me for a loop.

I finally got the correct answer, but I don't understand the solution in the solution manual.

(Image from Understanding sine at I can't add labels to the illustration, unfortunately.)

Look at the bottom side of the left-most triangle, the one with two overlapping triangles.

Assume that the red segment measures 6 cm, the green segment 4 cm.

I've been taught that you would find the scale factor using this equation:

6 x SF = 10

However, the solution manual shows:

6 x SF = 4

I started checking various right triangle problems to see whether you can find a correct scale factor this way....and I'm stumped.

Just based in the triangles I've looked at, the 6 x SF = 4 formula for the bottom red & green segments also holds true for the corresponding 6 and H3 segments of the hypotenuse:

6 x SF = H3

Obviously it does not hold true for the ratio between the two vertical sides labeled 2.6 and 3.9.

What's going on?


oh wow!

The page explains it!

hmmm . . .

I think Saxon blew it here. This was too big a leap for me inside a problem set.

Of course, Saxon isn't supposed to be a self-teaching book.

It may be time for me to take a class.

An actual class with an actual teacher.

Though I have to say, attempting to teach myself math I've never seen before is kind of cool.

My dad told me some relative of his taught himself calculus out of a book.

I like that idea.


Someone needs to write a sci-fi novel about homeschoolers preserving knowledge for the future.

Which reminds me, I'm still worried about the solution manual for Moise and Downs.

Once it's gone, then what?

Andrew persists

We got this email today from Clarice, Andrew's teacher:
Andrew has been pointing to a picture of the Arthur book Who is in Love with Arthur for a few days. Sometimes I think Andrew believes that we can make something appear just because he wants it. [ed: ditto]

Anyhow, yesterday he put the Arthur picture on his daily schedule and typed “dows lane Arthur bus yes.” We told him no, that we could not go on a bus to Dows Lane. So, while Annie was at lunch he came over to my desk with a pair of scissors and I asked him what he wanted. He pointed to my corkboard and I still did not know what he wanted. I told him to get what he wanted and he took my Irvington 2005-2006 directory off of the wall. On the front of it were tiny black and white pictures of each of the schools. He used the scissors to cut out the picture of Dows Lane, got a piece of Velcro off of the back of another item, and put it on the back of the Dows Lane picture. He then stuck the Dows Lane picture on his daily schedule next to the picture of the Arthur book. So, Annie went over to Dows Lane and got the book for him. He was happy for the rest of the afternoon and has been happy today until a 10 minute tantrum at noon. He typed “dows lane Arthur bus yes” again. I’d love to reinforce each effort to communicate, but the obsessive behavior that continually changes makes it very difficult.
This is a miracle.

Just a couple of years ago Andrew had no means of communication apart from grabbing our hands and pulling us to whatever it was that he wanted.

Now he's creating his own PECS cards.

He's still impossible; it's a wonder Clarice and Annie are still standing.

But they are.

Clarice and Annie persist, too.

so tell me, is this rote?

Educators love the false dilemma. One of their favored false dilemmas in math education is saying what they teach is "higher ordered thinking skills" and what was traditionally taught was "merely teaching by rote." Rote is not merely memorization, it is memorization without meaning or understanding. I contend that very little is taught by rote in any subject, even when memorization (or practice to automaticity) is required.

So let me give you an example of how one of the more difficult elementary math topics might be traditionally taught and you tell me if it's learning by rote.

Today's topic will be subtraction with regrouping (tens and ones). An example of such a problem is 66 - 37 = ?. Ordinarily, this topic gets taught after the student has learned how to do (and is firm on) subtraction without regroup (66 - 34 = ?). Let's further assume that the student knows how to do place value addition. This means the student knows how to decompose the number 66 into 60 + 6. In other words, the students knows that the number 66 comprises 6 tens and 6 ones.

Here's how the lesson might get taught traditionally:

Lesson One

I. Model Phase

When you work subtraction problems that use borrowing, you have to rewrite numerals so you have a new place-value addition. I'll show you how the new place value works.

[Write the number 36 on the board]

We're going to rewrite 36 for borrowing. We'll borrow 1 ten from the tens column and add that ten to the ones column.

How many tens do we start with? [point to the 3][students: 3]

I cross out the 3 and write the number that is 1 less than 3. What number is that? [students: 2]

Now I take the ten I borrowed and write it small in front of the 6.

The new place-value addition is 20 plus 16 equals 36.

We still have 36 because 20 plus 16 equals 36.

[repeat with a different number such as 57]

II. Lead Phase (if necessary)

Write the number 56 on the board.

Your turn. Cross out the 5 and write the number above it that is one less than 5. Then write the 1 ten you borrowed small in front of the 6. Raise your hand when finished.

(observe students and give feedback)

Check your work. Here's what you should have.

Everybody, say the new place-value addition for 56. [students: 40 plus 16 equals 56]

[repeat with another example if necessary]

III. Test Phase

[Write the numerals 84, 51, 45, and 72 on the board.]

Rewrite these numerals. Raise your hand when you're finished.

[Write on the board:]

Check your work. Here's what you should have.

Fix up any problems you got wrong.

End Lesson

[After the students are firm on the regrouping procedure, it's time to go on to using the procedure to solve subtraction problems]

Lesson Two

[Write on the board:]

You're going to do borrowing. For some column problems, you have to rewrite the top number so you can subtract. For other problems, you just subtract.

Here's how you figure out whether you need to borrow: You read the problem in the ones column. If the bottom number is bigger than the top number, you can't work the problem in that column, so you have to borrow.

Everybody, read the problem in the ones column. [students: 5 minus 5]

Can you work that problem? [students: yes]

So you don't have to borrow.

[Change the problem to:]

Can you work this problem? [students: No]

So you have to borrow.

[Repeat with a few more examples where borrowing is needed and not needed]

Give students a worksheet with the column subtraction problem: 53 -19 = ?

For this problem, you have to borrow because you can't work the problem in the ones column.

Rewrite the top number.

[Write on board:]

Check your work. Here's what you should have.

You'll make silly mistakes when you subtract unless you're careful about reading the new problem in the ones column.

I'll read the new problem in the ones column. 13 minus 9. That's a problem you can work.

You're going to work the problem now. Read the problem in the ones column. [students: 13 minus 9]. Read the problem in the tens column [students: 4 minus 1]

Write the answer to the problem. [check students work]

End Lesson

These lessons are taken from lessons 7 through 9 of Connecting Math Concepts, Level C which I've condensed a bit.

So tell me does anything in this lesson even remotely resemble rote learning?

The comments are open.

less math, more gambling

At least that's what they're recommending in Scotland:

Simple gambling games should be used as teaching tools in Scottish schools to make maths lessons more interesting, a leading academic said yesterday.

Professor Alastair Gillespie, chairman of the Scottish Mathematical Council, believes using dice and packs of cards in secondary school lessons would help pupils learn basic maths techniques such as probabilities. He also believes it would encourage more people to take up maths.

Too bad learning probability and statistics isn't the same as learning elementary math.

Wednesday, January 3, 2007

learning math is hard, part deux

(Part one of this post can be found here.)

In the first part of this post on the math program, Connecting Math Concepts, we were discussing about how the program is field tested and how error diagnosing and correction is built into the program. I needed to describe those two aspects of the program briefly to get to the aspect of the program that I intended to discuss -- practice.

Student practice is built right into CMC. That's one of the reasons why the program is field tested beforehand; to determine how much practice students need to retain the material taught. Unlike in most math programs, in CMC material is not just taught, tested and then permitted to lay fallow whereupon it is quickly forgotten by the student. Do you remember the threat of the dreaded end of year cumulative test back in K-12? You dreaded it because you knew that you had forgotten most of the material presented in the first half of the year. You don't have such a luxury in CMC, all tests are cumulative. The only time a skill isn't practiced or tested is because it's been incorporated into a more difficult skill. At the end of the year, students are expected to have retained all the material presented during the year. This is exactly what is needed when learning math.

Since CMC has been field tested with lower performing students and since CMC is designed to accelerate student learning as quickly as possible, you can get an idea for how much practice is needed for a lower performing student to retain the material. The program is designed to provide sufficient practice with a little bit extra to account for things like student absences, but not too much since that would hinder the acceleration. So, the practice provided in the program should turn out to be about what is necessary for a lower performing student to master the material at about the fastest rate he can handle. Cutting to the chase, the amount of practice that a lower performing student requires t o learn math is simply enormous if CMC is an accurate guide.

There is way too much practice for my son. I routinely cut out about every other practice lesson for each topic because I don't want him to get bored and our time for lessons is limited afterschool. Plus, I want to keep the ball rolling and stay far ahead of the wildly inappropriate nonsense that gets taught in his Everyday Math class.

So, you might be thinking that I'm only cutting out about half the material. Nope. I'm cutting out far more than that. I'm cutting out all the "extra practice" lessons that are scheduled for students after they fail a proficiency test. Since he's never failed any portion of any test so far, I haven't had to go back and reteach any lesson. At most he'll get a few problems wrong due to his desire not to being math work at night when he could be playing Lego Star Wars II on his PS2, but so far he's always stayed in the proficiency range no matter how fast I go.

In addition, I've never given him any worksheets from the extra practice workbooks or the blackline master worksheets. And, i skip all the games that sneak in more extra practice since there's no one to play against since he's the only student. Occasionaly, I'll play against him to give him an idea how fast I can work the problems so he has any idea how fast he's going to be expected to work the problems. He's not as fast as I am yet, but he routinely does his problems in half the time allotted in the timed exercises. So, he's starting to approach automaticity on some of the stuff he's learned so far.

Lastly, I've been known to skip the last 30 or lessons at the end of the year since most of this material will be quickly reviewed at the beginning at the next level.

I'd estimate that I cut out about 2/3 to 3/4 of the total practice provided in CMC which accords pretty closely with Engelmann's estimation that higher performers can be accelerated at about 3-4 times the rate of lower performers. And, it doesn't surprise me at all that lower performers need every last bit of all that practice I'm cutting out. Math is all about learning abstract concepts and our brains are not wired to learn abstract concepts easily. It also doesn't surprise me that in most math program, with the exception of Saxon, lower performing kids aren't getting close to the amount of practice they need to retain the math they've been taught.

Hence the widespread failure we see in math education.

And, this assessment doesn't even get into the messy area of the initial presentation of the material enabling the student to understand the concepts in the first place. I'll cover that aspect of CMC in future posts since we're now just starting to get into the interesting areas of math instruction. I'll leave you with this. CMC presents the material so clearly and concisely that I only have to "teach" for about five minutes each lesson. The rest of the time he's working problems using the skills I just taught or practicing previously taught skills. Ironically, that's probably far less teaching that goes on in your typical discovery learning/constructivist heavy math class. I'll show you who's the real guide on the side.

college prep

eduwonk links to The College Puzzle, a blog on college preparation that I suspect will become a regular read for me:
My blog discusses the important and complex subjects of college completion, college success, student risk factors (for failing), college readiness, academic preparation. I will explore the pieces of the puzzle that heavily influence, if not determine, college outcomes and success rates of college students. Furthermore, I've spent a great deal of time analyzing the messages that students receive about college preparation. I'll explore those messages and their roles in college outcomes.

Here are two of the factoids I've been looking for:

Only 22 percent of entering community-college students who want a four-year degree actually get one, nationwide. At minimally selective four-year colleges, fewer than half finish their degree. Too many students are not staying in college.

Ed said the other day that what we need to know about Irvington kids isn't how many go to college (practically everyone), but how many graduate from college in 4 to 6 years. I keep hearing about Irvington High School graduates who've dropped out of college after a year. I now know of at least 5 kids myself.

The plan is for the kids to go back, and I'm sure they will go back.

But it worries me that any "unplanned leaves" are happening at all.

In any case, affluent suburban schools should certainly be keeping track of their college graduation rate.

Author bio:
Michael W. Kirst is Professor Emeritus of Education and Business Administration at Stanford University since 1969.

Dr. Kirst received his Ph.D. in political economy and government from Harvard. Before joining the Stanford University faculty, Dr. Kirst held several positions with the federal government, including Staff Director of the U.S. Senate Subcommittee on Manpower, Employment and Poverty. He was a former president of the California State Board of Education.. His book From High School to College with Andrea Venezia was published by Jossey Bass in 2004.

Answers in the Toolbox

Answers in the Toolbox Revisited:

The report does show that of all eighth graders in 1988:

  • 78% graduated on time in 1992 with a standard diploma;
  • 53% entered postsecondary education directly from high school;
  • 48% persisted from their first to their second year of postsecondary study;
  • 35% earned a bachelor’s or associate degree by December 2000.

Ricky, my tutee

Like I said, I'm fairly new to pre-university education, though I see the results, and all my colleagues agree that we've been getting students with poorer and poorer math skills over the years. Having said that, we don't want to get into how I ended up tutoring (it's a long story). My tutee -- I'll call him Ricky -- is an 8th grader. His parents both have PhDs (before you ask why they aren't helping him, I do not know, and there really is no tactful way to ask). I'm being intentionally vague about details because, well, for personal and political reasons.

Ricky is extremely sharp, by the way. He immediately understands what I'm saying. I never have to explain anything more than once (he has the attention span of a gnat, though he is only in the 8th grade). Ricky enjoys learning, and is naturally curious. He's like a knowledge sponge. When I started working with him, he couldn't tell me what 16-9 was without reaching for his calculator. He had no mastery of basic operations, so that's what we did. He resisted at first (why do I have to do this when I can use my calculator?) but now that he has that information mastered, he can do things like basic algebra much more quickly and effortlessly, and he understand now.

Back to the course.

Okay, I may not have an ed PhD (thank God for that!) and I may not have pre-university classroom experience, but I've been teaching, writing exams, producing materials, and creating (and fiddling with) curricula for a long time. So even though I may not have any primary or secondary school teaching experience, the basic principles are the same.

They use worksheets (in-class worksheets -- the really frightening ones, with substance-free discussion and "food for thought" questions and "try this!" exercises -- as well as the take-home worksheets -- traditional math problems -- she gives them so they'll get decent scores on the state exams.

We have a problem right there. This is an admission that what they're doing in class -- the discussion and "food for thought" questions, and the "try this! Create three ways to calculate the area of a circle!" exercises -- are not teaching students the skills they need to get decent scores on the state math exams. If I were teaching the class and my students weren't learning the material, I'd change what I was doing. But this has not occurred to the teacher.

What strikes me, however, is how disorganized the course is.

They hop from topic to topic weekly, with no logical progression from topic to topic (for example, going from simple linear equations to probability -- and why they're doing probability in the 8th grade, I do not understand). They don't spend enough time on one topic to actually learn it, nor do they cover any topic to any depth, which makes no sense to me. The first thing on the list when they go back after break is limits -- why would 8th graders be doing limits when they don't understand basic fractions? Why would 8th graders be doing limits even if they did understand basic fractions?

The teacher also gives them "review" sheets (the sneer quotes are there because "review" implies a topic that's already been covered, and they don't cover a topic in any sense of the word). One was on long division. Ricky couldn't do it. He'd never seen anybody do long division, nor did he understand factoring. I asked him if his teacher had shown them how to do this, and he said no -- which normally I would take with a grain of salt, had I not seen so much disorganized nonsense already. He said she sent it home with this -- and he dug out another worksheet, a "how to" sheet on long division. He's in the 8th grade.

Forget the silly in-class worksheets, and forget the fact that the teacher sends them home with worksheets (traditional problems) so they'll get decent grades on the state exams. Here's my question: How can you send work home with your students and ask them to do something they've never been shown how to do? How can you justify that? If I tried to get away with that, I'd be in the Dean's office -- and it wouldn't be pretty.

The biggest battle has been teaching him to approach a problem, take it apart, and figure out how to solve it. It's been as much tutoring logic as math. (Does that mean I've been teaching "higher-order thinking" skills?) I had to be sort of ruthless at first because he didn't understand why he had to learn all this, but I've got him not only solving the problem, but checking it by finding a second way to do it and using that to work backwards. Like when we were doing probability (I still think that's weird). There was a basic problem on the worksheet, one of those standard balls problems (there are 30 red balls in the basket and the rest are blue. The probability of choosing a red ball is 0.75. How many balls are there total?) So I showed him how to calculate the total number of balls with the equation 0.75x=30. Then we calculated the number of blue balls, and checked it by first calculating the probability of choosing a blue ball, then subtracting that from one to find the probability of choosing a red ball. Once he started to catch onto the logic, he started picking it up fast. What's sad is that a kid that bright shouldn't need a tutor.

Another few months, and I'll turn Ricky into a card-carrying math geek.

on motivation

If you don't subscribe to the Direct Instruction listserv, you should.

Besides learning a lot of good stuff about good teaching techniques and other education issues, you get to witness an epic battle betwixt good and evil. In this case, evil is personified by Gerald Bracey who fancies himself as the gadfly of the board after joining a few months back. Problem is that Jerry is out of his league on the DI board as was demonstrated by this recent exchange on motivation.


Of course, no one mentions anything as soft and squishy as intrinsic motivation.


My comment on intrinsic motivation had to do with learning. Skinner could not account for his own behavior without it, although he tried.

Martin Kozloff (aka Professor Plum):

Intrinsic motivation is semantic nonsense.

Motivation. Motive force. Alleged to be "behind" or prior to action.

How do you know if a person "is" motivated?

They DO something.

WHY did they do something?

They were motivated.

Uh huh. Not too circular.

Extrinsic motivation. That is, the event towards which one is moving is external. Like what? A meal. She is cooking so that she can eat. He is training so that he will finish a race.

Yes, the event is outside, but why would anyone prepare a meal unless eating it felt good? Why would anyone run a race unless they anticipated a pleasing outcome.

In other words, the external event is NOT what motivates. It is merely the means to an end---which is feeling. Which is INTERNAL.

Therefore, so-called extrinsic motivation is really intrinsic. {Score one for me.]

If you say that intrinsic motivation might be wanting to do well, I'd like to know where you got the definition of "well." Was that intrinsic? Or was it gotten from the social environment?

And why would anyone want to do well? Because that is considered important in the culture. And where is this culture? It is OUTSIDE the person. So, intrinsic motivation is really extrinsic. [Score two for me.]

In summary, the words are kakos.

Finally, you don't need the concept of motivation to account for behavior---if by account you mean answering the question Why is she doing that.

The answer (if you include the concept of motivation) is that she is motivated to do so, which means nothing more than she wants to. Which is as helpful as explaining avoidance behavior by saying a person is afraid.

How do you know they are afraid?

They avoid the situation.

How come?

They are afraid.

How do you know?

They avoid the situation.

How come?


World without end, Amen.

[Score three for me. A hat trick.]

James MacDonald:

I, and possibly other members of the list, would like you to write complete
thoughts, including the evidence or rationale behind your comments. You
made a comment about Ohio State football, but did not say what the problems
were. You mention that Skinner could not account for his own behavior, but
do not give examples of this. Essentially, I am requesting a scholarly
discourse - not the out of the blue, non sequiturs you write.


I don't do pablum.


Mais non, mon ami, I must disagree.

Pablum is what you do best!

[Come on, young Bracey. That was funny, and you know it.]

To Be Continued?

weapons of math destruction

I learned of Weapons of Math Destruction from Linda Moran. (You can join Linda's listserv here.)

It's a hoot.

This is a good one.

learning math is hard

Update: For more one error detection and correction, take a look at this video (quicktime) starting at about 5:30. He's talks about error correction with reference to reading instruction. It continues on into this clip up until about 7:00 and math gets discussed for the last 3 minutes or so.

That's my current position based on teaching my six year old son math for the past year and a half.

Actually, that observation isn't based on my son having difficulty learning math. So far he hasn't. It's based on the the material we've skipped. It is that differential that separates the higher preforming math students from the lower performing math students. That differential represents an enormous amount of practice.

Unlike most parents who use Saxon to teach math, I'm using Connecting Math Concepts. Both programs are scripted, both use a mastery learning "basic skills" approach, and both have lots of practice built into the program. Both are complete programs which don't require parents to know how to teach math; knowing elementary math is sufficient. For most kids there is not much difference between the two. Contrast this with Singapore Math which does require some teaching skill to present and requires practice to be supplemented. That's not meant to be a knock against Singapore Math, each program has its strengths and weaknesses. I actually think that the ideal K-6 elementary math curriculum would be some combination of all three programs, capitalizing on the strengths of each.

For the purposes of this post, however, I want to focus on the practice aspect of learning math. To master elementary math a student needs to practice what's been learned until it is automatic. Unfortunately, most math programs do not provide sufficient practice to safeguard against the ravages of forgetfulness.

Most parents do not take control of the educational process until there the need to remediate becomes evident. At this point, there is a tension between the need to devote time for practice and the need to reteach the child to get him back on track as quickly as possible. Practice tends to get the short end of the stick at this point. It shouldn't.

One aspect I like about CMC is that it's been field tested so you can be certain that if the student has the math skills to enter a level of the program, the program will teach clearly enough and provide enough practice for the student to reliably master all the material presented in this level within one school year, about 120 lessons.

The most important aspect of CMC, however, is that error diagnosing and correcting are built right into the program, unlike almost every other math program. Let's face it, if students didn't make any errors while learning math, a trained monkey could teach math using almost any commercially available math program. It is in the diagnosing and correcting of student errors where most math programs fail. When students derail, many teachers are unable to get them back on the track. Math, being brutally cumulative is not forgiving at all when students derail.

This is CMC's greatest strength.

CMC is designed to minimize students errors in the first place by providing clear instruction in small instructional steps. Students are then tested frequently (workbooks are checked after every lesson and tests are given every two weeks) to check student errors. based on the ten unit tests, student errors are evaluated and a built-in remedy is provided to the student based on the errors the student made. The student is then retested to see if the remedy worked before the student is permitted to advance. If the student were permitted to advance without mastering the material, then the diagnosing and correction of errors would be become much more difficult come the next ten unit test because now the teacher doesn't know where the student went astray. Was it one of the new skills taught in the past ten lessons of was it one of the previously taught skills? Now extrapolate out 80 more lessons and try to figure out where the problem is for a newly taught skill that the student can't do. Forget about it.

Contrary to popular belief, the greatest shortcoming of the "constructivist" math programs is not the less than clear presentation of new skills, though this is certainly a problem; it is that error detection becomes virtually impossible. This is not so much a problem in a class full of higher performers, but it is deadly in a class where students make errors.

I see this post is getting a bit longish and I still haven't touched on the main point -- practice. So, I'm going to break it up into two posts since there's already much to chew on in this post. More to come.

Part two here

Andrew on the couch

Tuesday, January 2, 2007

math jokes

Via NYC Educator, I discovered Miss Cellania's math jokes.

Basic unit of laryngitis: 1 hoarsepower

365.25 days of drinking low-calorie beer: 1 lite year

1 million microphones: 1 megaphone

1 million aches: 1 megahurtz

1 millionth of a fish: 1 microfiche

It's subtle humor, of course.

SAT and sentence combining

I had the strongest feeling while taking the multiple choice sections of the SAT writing test that we need to commence sentence combining exercises around these parts.

Richard Hudson on Grammar teaching and writing skills: the research evidence

Grammar teaching could be surreptitious, as it were, with a clear underlying theory of grammar but minimal use of grammatical terminology. This is in fact how a lot of grammar teaching has been done; and in particular there is a well-recognised activity called 'sentence combining' which seems to be widely used in the USA. There is some evidence, apparently good, that this kind of activity benefits children's writing (Abrahamson 1977; Barton 1997; Hillocks 1986; Mellon 1969; O'Hare 1973), and in some studies it turned out that this kind of grammar teaching produced better results than more traditional teaching of grammatical analysis. For example, " Hillocks surveys the many studies of the effects of sentence combining, and finds them overwhelmingly POSITIVE at all levels (grade 2 to adult). 60% show significant gains in syntactic maturity; 30% non-significant gains; 10% no gains." (Weaver 1996, reporting Hillocks (1986)).

Why should these exercises be so much more successful than traditional analysis? It seems reasonable to assume that it is at least in part because they are exercises in the production of language, and specifically in the production of written language, so they feed much more directly into the child's growing repertoire of productive skills than exercises in grammatical analysis do. In short, they are more closely integrated into the teaching of writing, so the skills acquired in isolation are more likely to transfer directly into a usable skill. However this conclusion does not necessarily rule out the possibility of transfer from grammatical analysis under the right conditions.

This sentence combining quiz is pretty good. Here's a simple page for kids.

I think a couple of you may have rounded up sentence combining exercises the last time this come up. If so, I'd love to get the links.

(quick note: I didn't have good luck with Killgallon's books on sentence composing and neither did Susan S as I recall. I may go back to them in another year. We'll see.)

the trouble with fractions

Just found this at joannejacobs; Joanne found it at Chalkboard.

fractiongate ($)

no-knead bread

I made this today.

It tastes as good as it looks.

Costs approximately 20 cents per loaf.

There is nothing to it.

everything you need to know

As far as I can tell there is only one way to sc*** up this bread: forget to put the lid on the pan.

It's possible you can sc*** it up by adding 5x as much yeast as the recipe calls for. I don't know. I added 5x as much yeast as the recipe calls for to the same loaf I baked without a lid. (Must have the lid. Lid is key.)

You can not sc*** this up by mixing up a batch of dough, setting it out to rise, and then forgetting the whole thing until you happen to glance at your bake stand and think, Holy cow, how long has that been sitting there?

This may not sound like the definition of freedom, but it is. The fact that you can't sc*** this bread up by letting it rise too long means you don't have to get bogged down doing & re-doing mental math to figure out things like Where will I be in 18 hours when the dough finishes its first rise and Will I be asleep when the dough finishes its second rise?

None of that matters.

If you're some place else, it's not a problem.

If you're asleep, it's not a problem.

The dough can wait.

It's probably better if it does wait.


I'm relatively new to thinking about pre-university education -- my son has been out of school since before education went completely off the deep end, and had a relatively normal education. Not, mind, that I'm unfamiliar with ed schools and their faculty -- much the opposite, I fear.

I live in what is supposed to be one of the best school districts in the state, and have been tutoring an 8th grader in math. I may blog about it in the future, but right now, I'm too horrified.

That's enough. I have work to do. More later.

And thanks for inviting me!

toga! toga! toga!

Carolyn wrote,

"A group blog... this is going to be great!"

which I believe is a sly reference to an Animal House quote by Flounder who I'm thinking should be KTM's new mascot.

He was a legacy afterall.

Is math all that different than a toga party?

American Educator Magazine

My renewal form for American Educator just arrived, so I thought I'd tell everyone how to subscribe.

1 year $10
2 years $20
3 years $30

make check payable to:
American Federation of Teacheres
Order Department
555 New Jersey Avenue, NY
Washington, D.C. 2001

Money well spent.

SAT test

So I took the practice multiple choice test on writing.

I got 100% correct.

Thank God.

more later

sample SAT test
SAT test
get your recentered SAT scores right here

50 words or less

Barry asked us to come up with 50 words or less, then did it himself!
The NCTM has embodied the principles of constructivism, embedded them in their standards, and the textbooks that grew out of them (thanks to NSF funding) force classrooms to adhere to the constructivist non-think ethic whether they like it or not.
This reminds me of the description of UK writing instruction I keep coming back to:
[Judith] Koren describes how two British women she knows became effective essayists and speakers. “Each week, they’d had homework exercises like this: While preserving every essential point, reduce a 100-word essay to 50 words, then to 20, then to 10. Reduce 500 words to 50, 1,000 words to 100. Week after week, year after year....
(appeared in American Enterprise Magazine)

I love this from Woodrow Wilson, too:
If I am to speak for ten minutes, I need a week for preparation. If fifteen minutes, three days; if half an hour, two days; if an hour, I am ready now.

math is like a bowl of popcorn

From the Baltimore Sun:

But others disagree. John Haven, whose children attended Montgomery County public schools, believes far too much time in classrooms is spent on nonessential math work. He made an unscientific study of math teaching in Montgomery and concluded that, by his standards, 20 to 40 percent of what is taught is a waste of time. "It is a huge impact on math instruction. It is like having popcorn parties every Tuesday and Friday," he said.

I think that might have been John Hoven who gave us that quote. Anyway, read the whole thing.

Monday, January 1, 2007

back to the future

Thanks to Catherine for inviting me! A group blog... this is going to be great!

I would like to gather some thoughts from you all on what should happen to the original KTM website. I can leave it there indefinitely as long as we continue to pay the webhost, but should we really do that? There's a lot of great information there about mathematics and language arts resources and education -- think about the indexes and the threads and just plain great posts like Ken's response to the guy who flunked out of engineering -- but it can be very hard to dig through and find it. Should we parse through it and offer a condensed, informative website for people looking for information?

Or should we write an aggregate book -- with a title something like "Our math educations, ourselves"?

Or should we leave it alone?

constructivism -- just a hoax?

In an email from a noted education writer for a major metropolitan newspaper, this writer said the following with regard to constructivism:

Life is too short for me to try to distinguish for readers the difference between true constructivism and guided discovery. It really is not an interesting issue, because readers only care if their kids are learning, and I dont think I have ever seen anything in a public school that could qualify as true constructivism, and there is almost nobody out there with any influence pushing for t. c. It doesnt exist except in a very few private schools, if there.

This is from a writer who no matter what arguments you proffer in opposition, will respond, "oh, but what I meant was..." From what I've seen, the theory of constructivism has manifested itself in textbooks such as TERC's Investigations, or Connected Math Program, or Everyday Math, or IMP, or Core-Plus. One can make arguments that EM is not really constructivist. Well go ahead and make them. It sure isn't guided discovery. And it isn't mastery learning either. So you tell me what it is?

Adherents of constructivism will of say "Well of course we don't use constructivist techniques every day; it would be impractical." But they use them often enough. Giving students problems for which they do not have enough information or skills to solve them is but one example.

But enough of me talking. How would you respond to this darling of the edu-journalism community in 50 words or less?

January 1

Happy New Year!

calling Mark Roulo

I've gotten a bunch of invitations out, but am sure I'm still missing people.

One person I know I'm missing is Mark Roulo.

I need an email address!

labeling failure

From the Times:

To move forward, the country must also find new ways to support and transform failing schools, beyond labeling them failures and presuming that the stigma will inspire better performance.

I don't understand all the hub-bub over labeling schools that are failures as failures. There's nothing wrong with a little truth in advertising. Plus, there's the the what's good for the goose is good for the gander aspect. Schools have been labeling perfectly normal kids who've they failed to teach as "learning disabled" for quite some time now and I don't see the NYT bemoaning that state of affairs.

So what's wrong with calling these failing schools "teaching disabled"? No one ever complains when we call other failed businesses "bankrupt," "underperforming," or "struggling." Why should schools be immune from accurate labeling. At least when we label a school as failing it places the stigma back where it belongs--on the failed education professionals.

If we want the market to work like it's supposed to, accurate labeling is a necessity. When a new family is looking to move into the school district, we want the realtor to be able tell them that the school is "teaching disabled," so unless little Johnny and Susie are super students there's going to be lots or academic failure or the expense of tutors in the family's future.

Sunday, December 31, 2006

how it works

I see all kinds of people have joined - feel free to blog away!

fyi, you don't have to "join" in order to comment on posts.

sample SAT


I think the SAT math test is way harder today than it was back when I took it.

I've just this moment finished the 1st sample test in The Official SAT Study Guide. There were zillions of questions I had never seen before, nearly all of them having to do with functions.

(it wasn't zillions. it seemed like zillions.)

Obviously, I know nothing about functions.


I got 37 correct, 9 incorrect, for a raw score of 35, which translates to a range of 540 - 620.


If I'd been taking the test for real I would have scored somewhere between 600 - 650. I made "careless errors" I wouldn't have made with more adrenalin (and less autism-spiked cortisol) coursing through my system.

I also would have remembered to use my calculator.

Plus I would have used a test-taking strategy or two.

I completely forgot until I reached the 4th section that the plan with the SAT is:

  • do the easy questions first
  • use strategic guess-and-check throughout to eliminate obviously wrong answers

Instead I raced through the test, taking each problem as it came, and attempting to figure out how to solve it if even I'd never seen the thing before.


The good news is that I think it's fair to say I know Algebra 1 cold.

And I'm not bad on simple geometry.

I also know enough math (whatever that means) that I was able to figure out, on the fly, how to do a couple of problem types I'd never done before.

Nevertheless, the only way anyone's getting through this test is by dint of huge practice:

Section 3: 20 questions / 25 minutes
Section 6: 18 questions / 25 minutes (10 of the 18 are "student -produced response" - i.e., not multiple choice)
Section 9: 16 questions / 20 minutes

To do well on the SAT Math, you have to have reached automaticity for virtually every problem you see. There's no time to think or problem-solve; you have to be solving the question one second after you've finished reading the question.

That's an exaggeration, but not by much. I was able to scrape together a few minutes' thinking time left over from problems that took only a few seconds to do. But no more than a few.

Bottom line: the SAT is the exact opposite of everything the NCTM stands for.

Do we know how kids who've been in constructivist curricula in high school are doing on the SAT?


I've always thought SAT tutoring is a waste of money, which it may be (I've got some interesting studies to post).

However, after taking one practice test I can see where SAT tutoring for students who knew algebra 1, algebra 2, and geometry fairly well but hadn't practiced to automaticity could raise scores substantially.

Students definitely need to learn and practice test-taking strategies, too. No question.

That's another thing.

Back when I took the SAT, there were no practice tests. There wasn't even review.

You walked in, sat down, took the test.

I'd bet the ranch that the existence of the test prep industry has changed the tests.


I've completed Lesson 48 in Saxon Algebra 2, which is the second book in the "Saxon Math trilogy":

  • Saxon Algebra 1
  • Saxon Algebra 2
  • Advanced Mathematics
Geometry is integrated throughout the 3 books.

So: I have an SAT score in the low 600s after Saxon Algebra 1.

I think I'll take another sample test midway or perhaps 2/3 of the way through Algebra 2.


College Board list of math topics covered on SAT
sample SAT test
SAT test
get your recentered SAT scores right here

Barry's here!


hacking Blogger

Is there a simple way to customize Blogger templates?

I like this "skin," but the spacing between posts is confusing.

I think I need to do at least two things:

  • add white space between posts
  • reduce the font size on "posted by Catherine"

If anyone can explain this in 50 words or less, that would be great.

compare and contrast


....the top third of Singaporean students score, on average, in the top 10th percentile of students in the 38 countries tested on the TIMSS-R.

What the United States Can Learn From Singapore's World-Class Mathematics System
(and what Singapore can learn from the United States):
An Exploratory Study


...the very top top American kids are scoring about the 75th percentile on international studies. So we know our top performing kids are doing very well.

Mike McGill
New York state Superintendent of the Year

oldies but goodies:
more Singapore Math

Mike McGill


what is this?